Heil's minimum-volume conjecture for bodies of given thickness

Let $d$ be a prescribed thickness, and consider convex bodies in three-dimensional space with thickness $d$. The Heil body is the convex hull of six circular arcs of radius $d$, centered at the midpoints of the edges of a regular tetrahedron of edge length $d\sqrt{2}$, together with the four vertices of a rescaled tetrahedron of edge length $d(2\sqrt{6}-\sqrt{2})/3$. Heil's conjecture. The Heil body is the minimum-volume three-dimensional body in the class of convex bodies with given thickness. The conjecture concerns the unresolved minimum-volume problem for convex bodies of prescribed thickness; the source gives no resolution.